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Optical Anisotropy of Nanotube Suspensions Erik K JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 2 8 JULY 2004 Optical anisotropy of nanotube suspensions Erik K. Hobbiea) National Institute of Standards and Technology, Gaithersburg, Maryland 20899 ͑Received 16 January 2004; accepted 7 April 2004͒ A semimacroscopic model of an optically anisotropic nanotube suspension is derived perturbatively from Maxwell’s equations in a dielectric medium. We calculate leading-order expressions, valid in the dilute and semidilute limits, for the intrinsic and form contributions to the complex dielectric tensor in terms of the volume fraction, mean orientation, aspect ratio, optical anisotropy, and optical contrast of the nanotubes. The birefringence and dichroism are derived explicitly to leading order in fluctuations, and the connection with depolarized light scattering is established. The results are generalized to include tube deformation. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1760079͔ I. INTRODUCTION sions and melts. The approach is reminiscent of the general theory of flow-induced optical anisotropy in dilute polymer It is well known that flow can induce optical anisotropy solutions derived by Onuki and Doi,6 and it expands on a in macromolecular fluids and colloidal suspensions in the previous treatment of depolarized light scattering from car- 1,2 form of birefringence and dichroism. As first demonstrated bon nanotube suspensions.5 We derive general expressions 3 by Peterlin and Stewart, these effects separate naturally into for both the intrinsic and form contributions to the dielectric two distinct parts; an intrinsic contribution arising from the tensor, and we derive leading-order expressions for the bire- orientation of optically anisotropic polymer segments or par- fringence and dichroism. The connection to depolarized light ticles, and a form contribution arising from anisotropy in the scattering is established and the results are generalized to segmental or density correlation function. The experimental include tube deformation. techniques of flow birefringence and flow dichroism have proven to be powerful tools for studying the structural and dynamic properties of flowing polymer melts, polymer solu- II. INTRINSIC CONTRIBUTION tions, and anisotropic suspensions, which in turn has contrib- For a quasistatic suspension of N optically anisotropic uted to recent technological advances in the engineering and nanotubes in an optically isotropic solvent of dielectric con- ␧ flow processing of complex fluids. stant s illuminated by an incident electric field E(r,t) ϭ i(k rϪ␻t) 2ϭ␧ ␻2 2 Recently, carbon nanotubes have emerged as novel ma- E0e • , where k s /c and k points along the terials with potential applications in a host of advanced direction of propagation ͑see Fig. 1͒, the macroscopic dielec- 4 technologies. Nanocomposites engineered from polymers tric tensor can be defined via and carbon nanotubes, for example, offer the promise of ͑ ͒ϭ␧ ͑ ͒ϭ␧ i(k rϪ␻t) ͑ ͒ plastic materials with enhanced structural, thermal, elec- D r,t ˜ •E r,t ˜ •E0e • , 1 ␧ϭ͚3 ␧ tronic, and optical properties. Efficient bulk processing of where the tensor ˜ ␮,␯ϭ1xˆ␮ ␮␯xˆ␯ is homogeneous. If r such composites will depend in part on a detailed under- locates a segment of a nanotube, nˆ(r), then the spatially standing of the response of carbon nanotube suspensions and varying amplitude of the displacement field induced at r is melts to flow. As colloidal particles, nanotubes are also D͑r͒ϭ͕␣ʈnˆ͑r͒nˆ͑r͒ϩ␣Ќ͓1Ϫnˆ͑r͒nˆ͑r͔͖͒ E , ͑2͒ somewhat unique and intriguing, in that they straddle the • 0 interface between semiflexible polymers and microscopic fi- where the local complex dielectric constants ␣ʈ and ␣Ќ arise bers. Given the significant optical contrast between carbon from polarization and absorption along and normal to the nanotubes and most typical polymers, it is reasonable to ex- local symmetry axis of the nanotube, respectively, and 1 is pect that such suspensions will exhibit reasonably strong op- the identity tensor. The parameters ␣ʈ and ␣Ќ appear in the tical anisotropy, making them well-suited to rheo-optical formulas for the depolarized scattering of monochromatic techniques.5 Applied to such novel soft materials, the meth- light from nanotube suspensions.5 Equation ͑2͒ defines a ods of flow birefringence and flow dichroism might offer local dielectric tensor as new physical insight into both fundamental flow-structure ˜␧͑r͒ϭ␣ʈnˆ͑r͒nˆ͑r͒ϩ␣Ќ͓1Ϫnˆ͑r͒nˆ͑r͔͒, ͑3͒ relationships in anisotropic complex fluids, as well as funda- ␧ ϭ␧ mental information about the optical properties and optical when r falls within a tube and ˜ (r) s1 otherwise. Intro- Ϫ anisotropy of the nanotubes themselves. ducing the spatial average ͗˜␧͘ϭV 1͐dr˜␧(r), where V is the In this paper, we present a semimacroscopic calculation sample volume, we obtain the intrinsic part of the dielectric of the complex dielectric tensor of carbon nanotube suspen- tensor ␧ ϭ ␧ ϭ␾ ␣ ϩ␣ Ϫ ͒ ϩ Ϫ␾͒␧ ˜ 0 ͗˜͘ ͓ ʈ͗nˆ lnˆ l͘ Ќ͑1 ͗nˆ lnˆ l͘ ͔ ͑1 s1, a͒Electronic mail: [email protected] ͑4͒ 0021-9606/2004/121(2)/1029/9/$22.001029 © 2004 American Institute of Physics Downloaded 21 Jul 2004 to 129.6.154.32. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 1030 J. Chem. Phys., Vol. 121, No. 2, 8 July 2004 Erik K. Hobbie Multiplying by eϪik"r to remove the harmonic spatial depen- dence and taking the spatial average, this term averages to zero,8 and the first non-vanishing correction to D is thus second order in ␦˜␧(r), being ␦˜␧͑rЈ͒ ͵ Ј ͑ Ј͒ ЈÃ Јϫ ͵ Љ ͑ Ј Љ͒ ЉÃ Љ dr G r,r ␧ dr G r ,r “ “ s • “ “ FIG. 1. A cartoon showing an aligned nanotube suspension. The suspending ␧ medium is of uniform dielectric constant s . The incident field, E0(r,t), is ␦˜␧͑rЉ͒ linearly polarized along the ‘‘slow’’ axis of the suspension. After transmis- ϫ Љ ͑ ͒ ␧ D0 , 8 sion, the polarization is unchanged but the amplitude is attenuated by ab- s • sorption. An incident field linearly polarized along the ‘‘fast’’ axis, or at 90° to the shown scenario, is altered in a similar manner. Light linearly polarized which becomes at an angle between these directions will also emerge attenuated, but with a ␦˜␧͑rЈ͒ ␦˜␧͑rЉ͒ polarization that is elliptical. We seek to calculate the effective dielectric ͵ Ј ͵ Љ ik"rЉ ␧ ͓ Љ Љ dr ␧ dr e ␧ ˜ 0 E0 tensor of the suspension. s • s • • • “ “ ͒ ͑ Ϫٌ͑2͒Љ ͔ ͑ Ј Љ͒ ͓ Ј ЈϪٌ͑2͒Ј ͔ ͑ Ј͒ 1 G r ,r • “ “ 1 G r,r . 9 8 Ϫ Resolving the singularities in the Green’s functions, intro- where ␾ is the volume fraction and ͗nˆ nˆ ͘ϭN 1͚ nˆ nˆ .As l l l l l ducing their Fourier representation, multiplying by eϪik"r in a previous derivation of the light-scattering amplitude,5 we and taking the spatial average, the correction to ˜␧ is neglect contributions of linear and higher order in the differ- 0 ence between the end-to-end vector of the lth nanotube and ͗␦˜␧2͘ 1 ˜␧ ϭ ˜␧ ϩ ͵ dq˜T͑kϪq͒ ˜␧ its body director, nˆ l , which is equivalent to assuming that the 1 ␧2 • 0 ͑2␲͒3 • 0 tubes are nominally straight. The results will be generalized s to include deformation. Ϫqqϩk21 ͫ ͬ, ͑10͒ • q2Ϫ͑kϩi␵͒2 where the tensor III. SPATIAL FLUCTUATIONS ˜ ͑ ͒ϭ ͵ Ϫiq"r͗␦␧͑ ͒ ␦␧͑ ͒͘ ␧2 ͑ ͒ The earlier derivation gives just the leading-order part of T q dre ˜ r • ˜ 0 / s 11 the complex dielectric tensor. In general, there will be is related to the structure factor. Equation ͑10͒ is analogous higher-order contributions arising from the spatial variations to the expression derived by Onuki and Doi6 and is quite ␦˜␧(r)ϭ˜␧(r)Ϫ͗˜␧͘. Following Jackson’s treatment of Ray- general. As derived here, it contains two terms of leigh scattering,7 we obtain these terms perturbatively from O(␦˜␧2/␧2); a higher-order intrinsic correction arising purely the vector wave equation for D, which can be derived from s from mean-square fluctuations and a leading-order form cor- Maxwell’s equations in a dielectric medium. In the harmonic rection involving density correlations identical to that de- approximation in which all fields have a time dependence of ␻ rived in Ref. 6. This result is somewhat intuitive, as one the form ei t, this becomes an inhomogeneous vector Helm- could envision particles lacking any spatial correlation that holtz equation could still alter the polarization of incident light, with the 2ϩ 2͒ ϭϪ Ã Ã͑ Ϫ␧ ͒ ͑ ͒ physical scenario being optically anisotropic particles tooٌ͑ k D “ “ D sE . 5 small to visibly scatter. More quantitatively, for particles Following Jackson,7 the term in parenthesis on the right-hand ͗␦␧ ␦␧ ͘ lacking any segmental correlation ˜ (r)• ˜ (0) ͑ ͒ ␦␧ ␧ ϭ␧ 2 Ϫr2/2␰2 2 3 side of Eq. 5 is written as ˜ (r) D0 / s , where D0 ˜ 0 Ϸlim␰! ͗␦˜␧ ͘e ϰlim␰! ͗␦˜␧ ͘␰ ␦(r), and the second ik"r • 0 0 •E0e . As in scattering theory, a formal solution is then term in Eq. ͑10͒ vanishes, implying that for such ‘‘formless’’ obtained via successive iterations in terms of the Helmholtz ␧ nanotubes the leading-order intrinsic correction to ˜ 0 is sim- Green’s function ply proportional to the mean-square deviation.9 The full ex- ͉ Ϫ Ј͉ Ϫ Ј pression for the dielectric tensor is thus 1 eik r r 1 eiq•(r r ) ͑ Ј͒ϭ ϭ ͵ 2 G r,r 3 dq 2 2 , ͗␦˜␧ ͘ 1 4␲ ͉rϪrЈ͉ ͑2␲͒ q Ϫ͑kϩi␵͒ ␧Ϸ ϩ ϩ ͵ ˜ ͑ Ϫ ͒ ͑ ͒ ˜ ͭ 1 ␧2 ͑ ␲͒3 dqT k q 6 s 2 where the positive infinitesimal ␵ is needed to ensure causal- Ϫqqϩk21 6 ͫ ͬͮ ˜␧ ϩ..., ͑12͒ ity in the displacement field, D(r,t). The first-order correc- • q2Ϫ͑kϩi␵͒2 • 0 tion is ␧ where we have exploited that fact that ˜ 0 is symmetric.
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